The wave nature of electrons and the periodic lattice of atoms give rise to allowed energy bands and forbidden energy gaps for electrons in a solid. The forbidden gaps arise from the destructive interference of electrons for certain wavelengths and directions. If a forbidden gap exists for all possible directions, it is referred to as a complete bandgap. A semiconductor has a complete bandgap between the valence and conduction bands.
The optical analogy is the photonic crystal, where a periodic lattice of contrasting dielectric structures (i.e., different indices of refraction) provides the periodic potential for light that atoms do for electrons. Photonic crystals can be thought of as extensions of diffraction gratings (i.e., a one-dimensional photonic crystal) or naturally occurring crystals used in X-ray crystallography. Light interacting with a diffraction grating or X-ray crystal interacts with the periodic structure and is redistributed into “allowed” and precluded from “forbidden” directions. The forbidden directions are the “photonic bandgaps” of the structure.
Photonic crystals can be designed with photonic bandgaps that prevent light of a certain wavelength and direction from propagating within the photonic crystal. If the photonic crystal does not allow light to propagate within a wavelength range for all polarizations and directions, it is said to have a “complete photonic bandgap.” A necessary condition for a complete photonic bandgap is that the contrasting dielectric lattice be periodic in three dimensions (3D).
Research of photonic crystals and their behavior was prompted by the article by Yablonovitch, entitled “Inhibited spontaneous emission in solid-state physics and electronics,” in Phys. Rev. Lett. 58, No. 20, 2059–2062 (1987). Based on theoretical considerations, a number of new optical devices, from better lasers to extremely miniaturized light switches and guides, have been suggested by workers in this relatively new field.
While photonic crystals offer a great deal of promise in fabricating new devices, fabricating such crystals with predetermined structures is daunting. The article by Yablonovitch et al., entitled “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” in Phys. Rev. Lett. 67, No. 17, 2295–2298 (1991), describes the formation of the first artificial 3D photonic crystal by drilling an array of intersecting millimeter size holes in a dielectric material. This photonic crystal has a bandgap in the microwave range of the spectrum and is of limited practical interest.
Since the early pioneering work by Yablonovitch, a great deal of research has been devoted to the fabrication and study of photonic crystals in the infrared and visible. The article by Bimer et al., entitled “Silicon-based photonic crystals,” in Adv. Mater. 13, No. 6, Mar. 16, 2001, describes fabricating two-dimensional (2D) and 3D photonic crystals. 2D photonic crystals have periodicity in two dimensions and are uniform in the third dimension and are much easier to fabricate than 3D photonic crystals. Although a 2D photonic crystal can not have a complete bandgap in the strictest sense, it can have a forbidden gap that exists for all directions and polarizations of propagation precisely confined to the plane of periodicity. In this more limited sense, the forbidden gap is referred to as a “complete 2D bandgap.”
One application for a 3D photonic crystal having a complete bandgap is to guide light. This can be accomplished by carving a path into such a photonic crystal to serve as an air-filled waveguide. Light that propagates in the air-filled waveguide at a frequency within the complete bandgap will be totally reflected by the photonic crystal and be totally confined to and directed along the waveguide. It should confine light around tight bends much better than conventional waveguides (e.g., optical fibers), where the guiding depends on the limited angular range of total internal reflection at the interface between the higher index core and the lower index cladding.
Much work has been done in the area of 2D photonic crystals. For example, the formation of a two-dimensional array of very small cylindrical holes with a diameter of about 1 micron fabricated in a silicon substrate by electrochemical etching is describe in the article by Birner et al., entitled “Microporous silicon: A two-dimensional photonic bandgap material suitable for the near-infrared spectral range,” Phys. Status Solids, A 165, 111 (1998). As described in the article by Johnson et al., entitled “Guided modes in photonic crystal slabs,” Phys. Rev. B, 60 5751 (1999), this technique has been further developed to form a triangular lattice of 0.36 micron holes on a 0.5 micron pitch to produce a 2D photonic crystal with a “complete 2D bandgap” at a free space wavelength of 1.25 micron.
The article by Loncar et al., entitled “Waveguiding in planar photonic crystals,” Appl. Phys. Lett., Vol. 77, No. 13, 25 Sep. 2000, pp. 2813–2815, describes the fabrication of a 2D photonic crystal circuits designed and fabricated in silicon on silicon dioxide. The circuits include a planar waveguide that guides at 1.5 micron and utilizes a 2D photonic crystal consisting of a triangular lattice of cylindrical holes formed by chemically assisted ion-beam etching in silicon, as shown in FIG. 2 of the article. A silicon slab waveguide is formed by omitting one row of cylindrical holes from the 2D photonic crystal. The top and bottom surfaces of the slab waveguide and the photonic crystal are in contact with air. The structure utilizes 2D lateral confinement by the 2D photonic crystal, while confinement in the vertical (i.e., third dimension) is from conventional total internal reflection at the top and bottom Si/air interface. The article discusses propagation in straight sections and around 60° and 90° bends.
While 2D photonic crystal waveguides are useful for certain applications such as planar circuits and distributed feedback (DFB) lasers, there are a host of other applications (e.g., the formation of ultra-small optical and electro-optical integrated circuits and devices) that call for 3D photonic crystal waveguides. To date, however, readily forming 3D photonic crystal waveguides has proven difficult. This is particularly true where the desired bandgap wavelength is at the optical or infrared, since the dimensions of the lattice must be a fraction of the bandgap wavelength.
While some techniques have been developed for fabricating 3D photonic crystals, they involve extreme process conditions, such as forming individual dielectric layers and then stacking and bonding the layers to build the crystal. The formation of 3D waveguides in such crystals adds yet another level of complexity.
Accordingly, there is a need for an improved method of forming waveguides and waveguide-based devices from 3D photonic bandgap crystals.